In mathematics, a rational number is a number that can be written as the ratio of two integers, usually written as a / b, where b is not zero. Rational numbers are informally called fractions, but a whole number such as 5 is rational, since it can be written as 5/1. Numbers which are not rational are called irrational numbers.

Each rational number can be written in many forms, for example 3/6 = 2/4 = 1/2. The simplest form is when a and b have no common factors, and every rational number has a simplest form of this type. The decimal expansion of a rational number is either finite or eventually periodic, and this property characterises rational numbers. A real number that is not rational is called an irrational number.

Mathematically we may define them as an ordered pair of integers (a, b), with b not equal to zero. We can define addition and multiplication upon these pairs with the following rules:

(a, b) + (c, d) = (a × d + b × c, b × d)

(a, b) × (c, d) = (a × c, b × d)

To conform to our expectation that 2/4 = 1/2, we define an equivalence relation ~ upon these pairs with the following rule:

(a, b) ~ (c, d) if, and only if, a × d = b × c.

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense.

Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers. For instance, 5/7 = 1/2 + 1/6 + 1/21. For any positive rational number, there are infinitely many different such representations. These representations are called Egyptian fractions, because the ancient Egyptians used them. The hieroglyph used for this is the letter that looks like a mouth, which is transliterated R, so the above fraction would be written as R2R6R21. The Egyptians also had a different notation for dyadic fractions.

In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field: let p be a prime number and for any non-zero integer a let |a|p = p-n, where pn is the highest power of pdividinga; in addition write |0|p = 0. For any rational number a/b, we set |a/b|p = |a|p / |b|p. Then dp(x, y) = |x - y|p defines a metric on Q. The metric space (Q, dp) is not complete, and its completion is given by the p-adic numbers.